Tetrahedral manifolds and links Andrei Vesnin Sobolev Institute of Mathematics, Novosibirsk Second China-Russia Workshop on Knot Theory and Related Topics Novosibirsk, August 21– 25, 2015 A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 1 / 1
Outline: A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 2 / 1
Outline: 1. Cusped hyperbolic 3-manifolds. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 2 / 1
Outline: 1. Cusped hyperbolic 3-manifolds. 2. Tetrahedral manifolds. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 2 / 1
Outline: 1. Cusped hyperbolic 3-manifolds. 2. Tetrahedral manifolds. 3. Arithmeticity of tetrahedral manifolds. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 2 / 1
Outline: 1. Cusped hyperbolic 3-manifolds. 2. Tetrahedral manifolds. 3. Arithmeticity of tetrahedral manifolds. 4. Tetrahedral links. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 2 / 1
Outline: 1. Cusped hyperbolic 3-manifolds. 2. Tetrahedral manifolds. 3. Arithmeticity of tetrahedral manifolds. 4. Tetrahedral links. 5. Nice descriptions for infinite families. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 2 / 1
1. Cusped hyperbolic 3-manifolds. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 3 / 1
Cusped hyperbolic 3-manifolds A tetrahedron in H 3 is ideal if all its vertices belong to the absolute ∂ H 3 . A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 4 / 1
Cusped hyperbolic 3-manifolds A tetrahedron in H 3 is ideal if all its vertices belong to the absolute ∂ H 3 . Let M be a connected hyperbolic 3-manifold obtained by gluing together a finite set P of pairwise disjoint ideal tetrahedra. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 4 / 1
Cusped hyperbolic 3-manifolds A tetrahedron in H 3 is ideal if all its vertices belong to the absolute ∂ H 3 . Let M be a connected hyperbolic 3-manifold obtained by gluing together a finite set P of pairwise disjoint ideal tetrahedra. Let S be the set of all faces of tetrahedra from P . Assume that the gluing is realized by a pairing Θ along faces S by isometries of H 3 . A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 4 / 1
Cusped hyperbolic 3-manifolds A tetrahedron in H 3 is ideal if all its vertices belong to the absolute ∂ H 3 . Let M be a connected hyperbolic 3-manifold obtained by gluing together a finite set P of pairwise disjoint ideal tetrahedra. Let S be the set of all faces of tetrahedra from P . Assume that the gluing is realized by a pairing Θ along faces S by isometries of H 3 . The pairing Θ splits all ideal vertices in classes of equivalent. A class of equivalent ideal vertices is called a cusp of M . A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 4 / 1
Cusped hyperbolic 3-manifolds A tetrahedron in H 3 is ideal if all its vertices belong to the absolute ∂ H 3 . Let M be a connected hyperbolic 3-manifold obtained by gluing together a finite set P of pairwise disjoint ideal tetrahedra. Let S be the set of all faces of tetrahedra from P . Assume that the gluing is realized by a pairing Θ along faces S by isometries of H 3 . The pairing Θ splits all ideal vertices in classes of equivalent. A class of equivalent ideal vertices is called a cusp of M . Knot and link complements arise as examples of cusped manifolds. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 4 / 1
Complexity of cusped hyperbolic 3-manifolds. We say that complexity c ( M ) of a cusped hyperbolic 3-manifold M is equal to k if M admits an ideal triangulation with k tetrahedra and there is no an ideal triangulation with less number of tetrahedra. There is only a finite number of manifolds of a given complexity. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 5 / 1
Complexity of cusped hyperbolic 3-manifolds. We say that complexity c ( M ) of a cusped hyperbolic 3-manifold M is equal to k if M admits an ideal triangulation with k tetrahedra and there is no an ideal triangulation with less number of tetrahedra. There is only a finite number of manifolds of a given complexity. Problem. Classify cusped hyperbolic 3-manifolds. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 5 / 1
Complexity of cusped hyperbolic 3-manifolds. We say that complexity c ( M ) of a cusped hyperbolic 3-manifold M is equal to k if M admits an ideal triangulation with k tetrahedra and there is no an ideal triangulation with less number of tetrahedra. There is only a finite number of manifolds of a given complexity. Problem. Classify cusped hyperbolic 3-manifolds. Possible approach: Classify cusped hyperbolic 3-manifolds according to their complexity. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 5 / 1
Cusped hyperbolic manifolds of complexity � 8 . [P. Callahan – M. Hildebrand – J. Weeks, 1999] All 4 , 815 hyperbolic 3-manifolds which can be glued with � 7 ideal tetrahedra. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 6 / 1
Cusped hyperbolic manifolds of complexity � 8 . [P. Callahan – M. Hildebrand – J. Weeks, 1999] All 4 , 815 hyperbolic 3-manifolds which can be glued with � 7 ideal tetrahedra. [M. Thistlethwaite, 2010] All 12 , 846 hyperbolic 3-manifolds which can be glued with 8 ideal tetrahedra. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 6 / 1
Cusped hyperbolic manifolds of complexity � 8 . [P. Callahan – M. Hildebrand – J. Weeks, 1999] All 4 , 815 hyperbolic 3-manifolds which can be glued with � 7 ideal tetrahedra. [M. Thistlethwaite, 2010] All 12 , 846 hyperbolic 3-manifolds which can be glued with 8 ideal tetrahedra. [P. Callahan – J. Dean – J. Weeks, 1999],[A. Champanerkar – I. Kofman – T. Mullen, 2014] All hyperbolic knot complements which can be glued with at most 8 ideal tetrahedra. Tetrahedra 1 2 3 4 5 6 7 8 ≤ 8 1-Cusped Manifolds 0 2 9 52 223 913 3388 12241 16828 Knots 0 1 2 4 22 43 129 301 502 A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 6 / 1
Hyperbolic knot complements of small complexity. k=2: k=3: k=4: A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 7 / 1
2. Tetrahedral manifolds. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 8 / 1
Tetrahedral manifolds. We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tetrahedra. Let M be a tetrahedral manifold which can be decomposed into k regular ideal tetrahedra. Since regular ideal tetrahedron has maximal volume, c ( M ) = k. For k = 1 there is a unique tetrahedral manifold – H. Gieseking [1912], non-orientable. For k = 2 one of two orientable tetrahedral manifolds is the figure-eight knot complement. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 9 / 1
Tetrahedral manifolds which can be found in known tables. [P. Callahan, M. Hildebrand, J. Weeks]: listed all 4 , 815 hyperbolic 3-manifolds which can be glued from � 7 ideal (not necessary regular) tetrahedra. [M. Thistlethwaite]: listed all 12 , 846 hyperbolic 3-manifolds which can be glued from 8 ideal (not necessary regular) tetrahedra. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 10 / 1
Tetrahedral manifolds which can be found in known tables. [P. Callahan, M. Hildebrand, J. Weeks]: listed all 4 , 815 hyperbolic 3-manifolds which can be glued from � 7 ideal (not necessary regular) tetrahedra. [M. Thistlethwaite]: listed all 12 , 846 hyperbolic 3-manifolds which can be glued from 8 ideal (not necessary regular) tetrahedra. [Fominykh – Tarkaev – V., 2014]: independent generation of orientable tetrahedral manifolds of complexity at most 8 . Recognition: by homology and Turaev – Viro quantum invariants of 3-manifolds. Theorem. There are only 29 orientable tetrahedral manifolds of complexity at most 8 . Among them 17 have 1 cusp and 12 have 2 cusps. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 10 / 1
A census of tetrahedral manifolds. [E. Fominykh – S. Garoufalidis – M. Goerner – V. Tarkaev – V., arXiv:1502.00383] The list of all tetrahedral hyperbolic manifolds up to 25 tetrahedra for orientable case and up to 21 tetrahedra for non-orientable case is obtained. A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 11 / 1
A census of tetrahedral manifolds. [E. Fominykh – S. Garoufalidis – M. Goerner – V. Tarkaev – V., arXiv:1502.00383] The list of all tetrahedral hyperbolic manifolds up to 25 tetrahedra for orientable case and up to 21 tetrahedra for non-orientable case is obtained. joint with Evgeny Fominykh (Laboratory of Quantum Topology, Chelyabinsk, Russia) Stavros Garoufalidis (Georgia Institute of Technology, GA, USA) Matthias Goerner (Pixar Animation Studios, CA, USA) Vladimir Tarkaev (Laboratory of Quantum Topology, Chelyabinsk, Russia) A. Vesnin ( IM SB RAN ) Tetrahedral manifolds and links August 22, 2015 11 / 1
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